Bestvina–Brady [1]} applied this line of reasoning to piecewise Euclidean complexes in such a way that its parallels with Morse theory become apparent. Let \(K\)  be a piecewise Euclidean cell complex, and denote its vertex set by \(K^{(0)}\) . A function \(h : K \rightarrow \mathbb {R}\)  is called a discrete Morse function or a height function provided that the image \(h(K)\)  is a discrete subset of \(\mathbb {R}\)  and that \(h\)  is affine on closed cells and non-constant on edges.
Since all cells carry a Euclidean structure, we can think of links geometrically as spaces of directions. If a vertex \(v\)  occurs as a face of a cell \(c\) , the cell contributes a spherical piece to the link of \(v\)  consisting precisely of those directions based at \(v\)  pointing into \(c\) . The gluing maps for the complex \(K\)  induce gluings for those spherical pieces (each of which is a spherical polyhedron with faces coming from cells between \(v\)  and \(c\) ). The descending link of \(v\)  consists of the contributions of those proper cofaces \(c\)  such that \(h\)  attains its maximum on \(c\)  at \(v\) . The ascending link of \(v\)  comes from those proper cofaces such that \(h\)  attains its minimum on \(c\)  at \(v\) .
